## Fluctuations in quantum mechanics and field theories, with A. Turbiner (UNAM) and E. Shuryak (Stony Brooke)

Semiclassical approximations are well known tools, both in quantum mechanics and quantum field theory. Standard textbooks of quantum mechanics usually start with the Bohr-Sommerfeld quantization conditions and the semiclassical WKB approximation for the wave function. Unfortunately, extending such methods beyond the first correction for the one-dimensional case or those with separable variables, or to a multidimensional case, proved to be difficult. In mathematical physics, general constructions for multidimensional complexified saddle points are related to their extrema and the network of the so-called Lefschetz thimbles (solutions of the gradient flow equations) connecting them. Applications of such a theory in the path-integral context have been investigated by Witten, which inspired a number of subsequent studies. Semiclassical methods that start with Feynman path integrals, on the other hand, are applicable not only for many dimensions or many-body applications, but even for quantum field theories. The so-called instanton calculus has been developed for gauge theories and other models. Instantons are indeed responsible for many important phenomena in quantum field theories (QFTs). From a technical point of view, however, so far the calculations have not progressed beyond one loop in any of these QFT applications.